algebraic expressions & polynomials chapter 5 sections 5.1-5.3

Algebraic Expressions & Polynomials

Chapter 5 Sections 5.1-5.3

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Fundamental Operations5.1

• In arithmetic, we perform mathematical operations with specific

numbers.

• In algebra, we perform these same basic mathematical operations

with numbers and variables—letters that represent unknown

quantities.

• To begin our study of algebra, some basic mathematical principles

that you will apply are listed below.

• Note that “≠” means “is not equal to.”

Fundamental Operations

• Basic Mathematical Principles

• 1. a + b = b + a (Commutative Property for Addition)

• 2. ab = ba (Commutative Property for Multiplication)

• 3. (a + b) + c = a + (b + c) (Associative Property for Addition)

• 4. (ab)c = a(bc) (Associative Property for Multiplication)

• 5. a(b + c) = ab + ac, or (b + c)a = ba + ca (Distributive Property)


• 6. a + 0 = a

• 7. a 0 = 0

• 8. a + (–a) = 0 (Additive Inverse)

• 9. a 1 = a

• 10. a = 1 (a ≠ 0) (Multiplicative Inverse)


• In mathematics, letters are often used to represent numbers.

• Thus, it is necessary to know how to indicate arithmetic operations

and carry them out using letters.

• Addition: x + y means add x and y.

• Subtraction: x – y means subtract y from x or add the

negative of y to x; that is, x + (–y).

• Multiplication: xy or x y or (x)(y) or (x)y or x(y) means

multiply x by y.


• Division: x y or means divide x by y, or find a

number z such that zy = x.

• Exponents: xxxx means use x as a factor 4 times, which is

abbreviated by writing x4.

• In the expression x4, x is called the base, and

4 is called the exponent.

• For example, 24 means 2 2 2 2 = 16.


• Order of Operations

• 1. Perform all operations inside parentheses first. If the problem contains a fraction bar, treat the numerator and the denominator separately.

• 2. Evaluate all powers, if any. For example, 6 23 = 6 8 = 48.

• 3. Perform any multiplications or divisions in order, from left to right.

• 4. Do any additions or subtractions in order, from left to right.


• Evaluate: 4 – 9(6 + 3) (–3).

• = 4 – 9(9) (–3)

• = 4 – 81 (–3)

• = 4 – (–27)

• = 31

Example 1

Add within parentheses.

Multiply.

Divide.

Subtract.

• To evaluate an expression, replace the letters

with given numbers; then do the arithmetic using

the order of operations.

• The result is the value of the expression.

• Evaluate ab/3c + c if a=6 b=10 c= -5

• 6(10)/3(-5) + (-5)= 60/-15 + (-5)= -4 + (-5)=

• -9



Simplifying Algebraic Expressions5.2

• Parentheses are often used to clarify the order of

operations when the order of operations is complicated

or may be ambiguous.

• Sometimes it is easier to simplify such an expression by

first removing the parentheses—before doing the

indicated operations.

Simplifying Algebraic Expressions

• Two rules for removing parentheses are as follows

• Removing Parentheses

• 1. Parentheses preceded by a plus sign may be removed without changing the signs of the terms within. Think of using the Distributive Property, a(b + c) = ab + ac, and

multiplying each term inside the parentheses by 1. That is,

• 3w + (4x + y) = 3w + 4x + y


• 2. Parentheses preceded by a minus sign may be removed

if the signs of all the terms within the parentheses are

changed; then the minus sign that preceded the

parentheses is dropped. Think of using the Distributive

Property, a(b + c) = ab + ac, and multiplying each term

inside the parentheses by –1. That is,

• 3w – (4x – y) = 3w – 4x + y

• (Notice that the sign of the term 4x inside the

parentheses is not written. It is therefore understood to

be plus.)


• Remove the parentheses from the expression

• 5x – (– 3y + 2z).

• 5x – (– 3y + 2z) = 5x + 3y – 2z

Example 1

Change the signs of all of the terms within parentheses; then drop the minus sign that precedes the parentheses.

• A term is a single number or a product of a number and one or more letters raised to powers. The following are examples of terms:

• 5x, 8x2, – 4y, 15, 3a2b3, t

• The numerical coefficient is the numerical factor of a term.

• The numerical factor of the term 16x2 is 16.• The numerical coefficient of the term – 6a2b is – 6.• The numerical coefficient of y is 1.


• Terms are parts of an algebraic expression separated by

plus and minus signs.

• For example, 3xy + 2y + 8x2 is an expression consisting

of three terms.


Like Terms

• Terms with the same variables with exactly the same exponents are called like terms.

• For example, 4x and 11x have the same variables and are like terms.

• The terms – 5x2y3 and 8x2y3 have the same variables with the same exponents and are like terms.

• The terms 8m and 5n have different variables, and the terms 7x2 and 4x3 have different exponents, so these are unlike terms.

Like Terms

• The following table gives examples of like terms and unlike terms.

Example 3

• Like terms that occur in a single

expression can be combined into one term

by combining coefficients (using the

Distributive Property).

• Thus,

• ba + ca = (b + c)a.

Like Terms

• Combine the like terms 2x + 3x.

• 2x + 3x = (2 + 3)x

• = 5x

Example 4

• Some expressions contain parentheses that must be removed before combining like terms. Follow the order of operations.

•

• a(b + c) = ab + ac.

• The Distributive Property is applied to remove

parentheses when a number, a letter, or some product

precedes the parentheses.

Like Terms

• Simplify: 3x + 5(x – 3).

• 3x + 5(x – 3) = 3x + 5x – 15

• = 8x – 15

Example 12Apply the Distributive Property by multiplying each term within the parentheses by 5.

Combine like terms.


Addition and Subtraction of Polynomials5.3

• A monomial, or term, is any algebraic expression that contains only

products of numbers and variables, which have nonnegative integer

exponents.

• The following expressions are examples of monomials:

• 2x, 5, –3b,

• A polynomial is either a monomial or the sum or difference of

unlike monomials. We consider two special types of polynomials.

Addition and Subtraction of Polynomials

• A binomial is a polynomial that is the sum or difference of two

unlike monomials. A trinomial is the sum or difference of three

unlike monomials.

• The following table shows examples of monomials, binomials, and

trinomials.


• Expressions that contain variables in the denominator are not polynomials.

• For example,

• and

• are not polynomials.


• Find the degree of each monomial:

• a. –7m, b. 6x2, c. 5y3, d. 5.

• a. –7m has degree 1.

• b. 6x2 has degree 2.

• c. 5y3 has degree 3.

• d. 5 has degree 0

Example 1

The exponent of m is 1.

The exponent of x is 2.

The exponent of y is 3.

5 may be written as 5x0.

• A polynomial is in decreasing order if each term is of some degree less than the preceding term.

• The following polynomial is written in decreasing order:

• 4x5 – 3x4 – 4x2 – x + 5


exponents decrease

• A polynomial is in increasing order if each term is of some degree larger than the preceding term.

• The following polynomial is written in increasing order:

• 5 – x – 4x2 – 3x4 + 4x5

• Adding Polynomials• To add polynomials, add their like terms.


exponents increase

• Add: (3x + 4) + (5x – 7).

• (3x + 4) + (5x – 7) = (3x + 5x) + [4 + (–7)]

• = 8x – 3

Example 3

Add the like terms.

• Subtracting Polynomials

• To subtract two polynomials, change all

the signs of the terms of the second

polynomial and then add the two resulting

polynomials.


• Subtract: (5a – 9b) – (2a – 4b).

• (5a – 9b) – (2a – 4b)

• = (5a – 9b) + (–2a + 4b)

• = [5a + (–2a)] + [(–9b) + 4b]

• = 3a – 5b

Example 7

Add the like terms.

Change all the signs ofthe terms of the secondpolynomial and add.

Group Practice Problems

•Page 241

•3, 4, 5, 7

algebraic expressions & polynomials chapter 5 sections 5.1-5.3

Documents

x y means

x y means

negative of y

ab c

arithmetic operations

b c associative property

basic mathematical operations

indicated operations